
where \(\mathbf {s}_{k,l}\in \wp \subset \mathbb {C}^{\tau }\)  is the transmitted sequence of complex valued baseband symbols from the \(k\) th device over \(\tau \) 
channel uses. \(\mathbf {W}_l \in \mathbb {C}^{M\times \tau }\)  is the additive white Gaussian noise. Herein,
the coefficient \([\mathbf {v}_l]_n=e^{\jmath \theta _{n}}, n\in \lbrace 1, \cdots , N\rbrace \)  denotes the phase shift of RIS reflecting element \(n\)  [1]}. \(\theta _{n}\in [0, 2\pi ]\)  and phase shift vector \(\mathbf {v}_l\)  is constant during the \(l\) -th subblock and varies from one subblock to another subblock.
Let \(\mathbf {y}_l=\mathrm {vec}(\mathbf {Y}_l)\)  and \(\mathbf {w}_l=\mathrm {vec}(\mathbf {W}_l)\)  denote the vectorized versions of \(\mathbf {Y}_l\in \mathbb {C}^{M\times \tau }\)  and \(\mathbf {W}_l\) , respectively. Moreover, we rearrange \(\hat{\mathbf {U}}\text{diag}(\mathbf {A}_{\mathrm {R}}\lambda _k)\mathbf {v}_l\)  as
\(\hat{\mathbf {U}}\text{diag}(\mathbf {A}_{\mathrm {R}}\lambda _k)\mathbf {v}_l\!\!=\!\!\sum _{n=1}^{N}\!\hat{\mathbf {U}}(:,n)\lbrace \mathbf {A}_{\mathrm {R}}\lambda _k\rbrace  (n)\mathbf {v}_l(n)\!\!=\!\!\hat{\mathbf {V}}_l\mathbf {A}_{\mathrm {R}}\lambda _k,\) 
where \(\hat{\mathbf {V}}_l=[\mathbf {v}_l(1)\hat{\mathbf {U}}(:,1),\mathbf {v}_l(2)\hat{\mathbf {U}}(:,2),\cdots ,\mathbf {v}_l(N)\hat{\mathbf {U}}(:,N)]\in \mathbb {C}^{M\times N}\) . Hence, (REF ) can be equivalently rewritten as
\(\mathbf {y}_l=\sum _{k=1}^{K_a}\mathbf {s}_{k,l}\otimes \mathbf {P}_l\lambda _k+\mathbf {w}_l,\) 
